A very efficient and “unreasonably effective” [1]} tool for the computation of homotopy colimits in simplicial model categories is given by the two-sided (simplicial) bar construction.
We recommend the book [1]} as an introduction and as a reference.
Let \({I}\)  be a small category, and consider \({I}\) -shaped diagrams in a simplicial category \({{M}}\) .
Heuristically, the homotopical meaningfulness of the bar construction stems from the fact that it introduces coherence data into diagrams \(D \colon {I}\rightarrow {{M}}\)  by keeping track of composable sequences of morphisms in \({I}\) .
That is, it takes into account all \(n\) -simplices of the nerve \(N{I}\) .
